Đặt: $t=\ln x \Rightarrow dt=\dfrac{1}{x}dx$
Ta có:
$\int\limits_1^e\dfrac{\ln x}{x(\ln x+2)^2}dx$
$=\int\limits_0^1\dfrac{t}{(t+2)^2}dt$
$=\int\limits_0^1\left(\dfrac{1}{t+2}-\dfrac{2}{(t+2)^2}\right)dt$
$=\left(\ln(t+2)+\dfrac{2}{t+2}\right)\left|\begin{array}{l}1\\0\end{array}\right.$
$=\ln\dfrac{3}{2}-\dfrac{1}{3}$